Conditional propagation of chaos for mean field systems of interacting neurons

TL;DR

This paper proves that a system of interacting neurons converges to a limit described by a nonlinear stochastic differential equation with common noise, establishing a conditional propagation of chaos property.

Contribution

It introduces a new framework for analyzing mean field neuron systems with common noise, proving convergence and well-posedness of the limit equation.

Findings

Convergence of finite neuron system to a nonlinear stochastic differential equation.

Establishment of conditional independence of neurons given the common noise.

Introduction of a new martingale problem suited for systems with common noise.

Abstract

We study the stochastic system of interacting neurons introduced in De Masi et al. (2015) and in Fournier and L\"ocherbach (2016) in a diffusive scaling. The system consists of $N$ neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to $0$ and all other neurons receive an additional amount of potential which is a centred random variable of order $ 1 / \sqrt{N}.$ In between successive spikes, each neuron's potential follows a deterministic flow. We prove the convergence of the system, as $N \to \infty$, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion $W$ which is created by the central limit theorem. This Brownian motion is underlying each particle's motion and induces a common noise factor for all neurons in the limit system. Conditionally on $W,$ the different neurons are independent in the limit system. This is the {\it conditional propagation of chaos} property. We prove the well-posedness of the limit equation by adapting the ideas of Graham (1992) to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.